text:
\section{Introduction}
\label{sec:intro}

Topological solitons
are ubiquitous in nature,
playing significant roles in
quantum field theories
\cite{Rajaraman:1987,Rubakov:2002fi,Manton:2004tk, Shnir:2005vvi,Vachaspati:2006zz,Dunajski:2010zz,Weinberg:2012pjx,Shnir:2018yzp},
supersymmetric gauge theories
\cite{Tong:2005un,Tong:2008qd,Eto:2006pg,Shifman:2007ce,Shifman:2009zz},
QCD
\cite{Eto:2013hoa},
cosmology
\cite{Kibble:1976sj,Kibble:1980mv,Vilenkin:1984ib,Hindmarsh:1994re,Vachaspati:2015cma,Vilenkin:2000jqa},
and various condensed matter systems
\cite{Mermin:1979zz}
such as
helium-3 superfluids \cite{Volovik:2003fe},
superfluids \cite{Svistunov:2015},
superconductors
\cite{Blatter:1994zz,Giamarchi:2002},
Josephson junctions of superconductors \cite{Ustinov2015},
Bose-Einstein condensates (BECs) of ultracold atomic gasses
\cite{Kawaguchi:2012ii},
nonlinear media
\cite{Pismen,Bunkov:2000},
liquid crystals \cite{Pismen}
and magnets \cite{Nagaosa2013,Gobel:2020mqd}.
Among various salient features of topological solitons,
formations of topological solitons are
one of the most important aspects.
Topological solitons are known to be created
at phase transitions accompanied with
a spontaneous symmetry breaking (SSB).
This is known as the Kibble-Zurek mechanism
\cite{Kibble:1976sj,Kibble:1980mv,Zurek:1985qw,Zurek:1996sj}
and was experimentally confirmed in various condensed matter systems
such as
liquid crystals \cite{Bowick:1992rz},
helium-4 superfluids \cite{Hendry:1994},
helium-3 superfluids \cite{Baeuerle:1996zz,Ruutu:1995qz},
and ultracold atomic gasses
\cite{Sadler:2006,Weiler:2008}.
In two spatial dimensions, a pair of a vortex and an anti-vortex
is created at finite temperature, which is known as
Berezinskii-Kosterlitz-Thouless transition (BKT) transitions
\cite{Berezinsky:1970fr,Berezinsky:1970-2,Kosterlitz:1972,Kosterlitz:1973xp}
(see also Refs.~\cite{Kobayashi:2018ezm,Kobayashi:2019sus}
for recent studies).
This was  experimentally confirmed in thin-film
helium-4 superfluids \cite{PhysRevLett.40.1727},
superconductors \cite{PhysRevLett.42.1165},
and ultracold atomic gases \cite{Hadzibabic:2006}.
There are several other mechanisms for creating topological solitons
such as
bubble collisions \cite{Hawking:1982ga,Copeland:1996jz,Digal:1995vd},
monopole pair creation under strong magnetic field
(as a dual to the Schwinger mechanism)
\cite{Affleck:1981ag, Affleck:1981bma},
and brane annihilations
\cite{Sen:2004nf}
(see
Refs.~\cite{Bradley:2008zza,Takeuchi:2012ee,Nitta:2012hy,Takeuchi:2012ap}
for condensed matter analogues).

The purpose of this paper is to propose a yet another mechanism for
a creation of topological solitons, that is,
quantum nucleation through quantum tunneling.
This mechanism works when
the ground state is ``solitonic''.
When the Lagrangian or Hamiltonian
contains a certain type of
a topological term
with its coefficient
 larger than a certain critical value,
the energy of topological solitons is negative
and thus they are spontaneously created in uniform states.
However, one cannot place infinite number of solitons since
they repel each other, and thus
the ground state is a lattice of topological solitons.
A typical example of solitonic ground states is given by
chiral soliton lattices (CSLs)
which are periodic arrays of domain walls or solitons,
appearing
in various condensed matter systems:
cholesteric liquid crystals \cite{DEGENNES1968163}
and
chiral magnets \cite
togawa2012chiral,togawa2016symmetry,KISHINE20151,
PhysRevB.97.184303,PhysRevB.65.064433,Ross:2020orc}
with
the Dzyaloshinskii-Moriya (DM) interaction \cite{Dzyaloshinskii,Moriya:1960zz}.
The latter has an important nanotechnological application
in information processing such as magnetic memory storage devices and magnetic sensors \cite{togawa2016symmetry}.
The $O(3)$ sigma model together with the DM term
reduces to
the sine-Gordon model plus a topological term at low energy,
and the CSL is a sine-Gordon lattice.
Another condensed matter example of solitonic ground states is
given by
magnetic skyrmions
\cite{Bogdanov:1989,Bogdanov:1995}
in chiral magnets,
which typically constitute
 a triangular lattice in the ground state
 in the parameter region in which the DM term is strong enough
\cite{Rossler:2006,Han:2010by,Lin:2014ada,Ross:2020hsw}.
Since they have been realized in laboratory experiments
\cite{doi:10.1126/science.1166767,doi:10.1038/nature09124},
there has been great interests such as an application to
 information carriers in ultradense memory and logic devices
  with low energy consumption
\cite{doi:10.1038/nnano.2013.29}.
The other examples are for instance
${\mathbb C}P^{N-1}$ skyrmion lattices in
$SU(N)$ magnets
\cite{Akagi:2021dpk, Akagi:2021lva, Zhang:2022,
Amari:2022boe} and 3D skyrmions
\cite{Kawakami:2012zw}
in spin-orbit coupled BECs
with background gauge fields as generalizations of the DM term.

Recently, it has been predicted
that CSLs are also ground states
of QCD at finite density under
strong magnetic field
\cite{Son:2007ny,Eto:2012qd,Brauner:2016pko,Chen:2021vou,Gronli:2022cri}
or under rapid rotation
\cite{Huang:2017pqe,Nishimura:2020odq,Eto:2021gyy},
due to a topological term originated from
the chiral magnetic effect (CME)
\cite{Son:2004tq,Son:2007ny}
which is the vector current in the direction of the magnetic field,
or chiral vortical effect
(CVE) \cite{Vilenkin:1979ui,Vilenkin:1980zv,
Son:2009tf} which is the axial vector current in the direction of the rotation axis, respectively.
They also appear with thermal fluctuation \cite{Brauner:2017uiu,Brauner:2017mui,Brauner:2021sci}
(see also Refs.~\cite{Yamada:2021jhy,Brauner:2019aid,Brauner:2019rjg}).
In the CSLs,
the number density of solitons is determined by
the strength of external fields such as
a magnetic field or rotation
(or the DM term for chiral magnets).
As external fields are larger
 above the critical value,
the soliton number density is larger.
Thus, when one gradually increases(decreases)
the strength of the external field,
the mean inter-soliton distance decreases(increases) accordingly.
One of natural questions is
how they are created from the vacuum (uniform state).
When one instantaneously changes
the external field from the value below the critical value
to the one above the critical value,
it is unnatural that a flat soliton (domain wall)
with infinite world-volume instantly appears.
Instead, quantum nucleation can occur in this case
as we propose in this paper.

To explain our mechanism, it is worth to recall
quantum decay of a metastable false vacuum and bubble nucleation
first formulated by Coleman
\cite{Coleman:1977py,Callan:1977pt,Coleman:1987rm}
(see Refs.~\cite{Rubakov:2002fi,Weinberg:2012pjx,Coleman:1985rnk}
 as a review).
Decay probabilities can be calculated by
evaluating the Euclidean action values for
bounce solutions.
In the thin-wall approximation, one can evaluate
the decay probability in terms of tensions of domain walls.
Preskill and Vilenkin studied
quantum decays of metastable topological defects
\cite{Preskill:1992ck}
(see Ref.~\cite{Vilenkin:2000jqa} as a review,
and Refs.~\cite{Monin:2008mp,Monin:2008xx,Monin:2009gi}
for recent studies).
One of typical cases is given by an axion model,
in which a domain wall (or soliton) terminates on a string.
Thus, a domain wall is metastable and
can decay by quantum tunneling with creating a hole
bounded by a closed string.
Again in the thin-wall approximation, one can evaluate
the decay probability of the domain wall
in terms of tensions of domain walls and strings.
Some examples are given by domain walls in two-Higgs doublet models
\cite{Eto:2018hhg,Eto:2018tnk}
and axial domain wall-vortex composites
in QCD \cite{Eto:2013bxa}.
Another case is a string (vortex) ending on a monopole.
In this case, a string is metastable  and
decays by cutting the string into two pieces
whose endpoints are attached by a monopole and an anti-monopole
through quantum tunneling.
Examples
can be found
for instance for
electroweak $Z$-strings in the standard model
\cite{Nambu:1977ag,Achucarro:1999it,Eto:2012kb} and
non-Abelian strings in dense QCD \cite{Vinci:2012mc,Eto:2013hoa}.

In this paper,
we study quantum nucleation of topological solitons
through quantum tunneling.
For definiteness,
we discuss chiral solitons in a complex $\phi^4$ model
(an axion model with the domain wall number one) with a topological term,
which is a simplification of  low-energy theories of
chiral magnets (with an easy-plane anisotropy)
and QCD at finite density
under strong magnetic field or rapid rotation.
The origin of the topological term is
 the DM interaction for chiral magnats, while it is
CME and CVE for QCD under strong magnetic field or rapid rotation,
respectively.
If the external field $B$ is larger than a certain critical value
$B_c$, the soliton tension is
effectively negative, and therefore it can be created by quantum tunneling.
We estimate the nucleation probability analytically
in the thin-defect approximation in any dimension,
and fully calculate it numerically
in $2+1$ and $3+1$ dimensions by
using the relaxation (gradient flow) method.
In $2+1$ dimensions, a vortex is particle-like,
a soliton is string-like,  and
a pair of a vortex and an anti-vortex is connected
by a linear soliton,
while
in $3+1$ dimensions, a vortex is string-like, a soliton is wall-like,
and a disk of a soliton wall is bounded by a string loop.
Once such a composite configuration of a finite size is created
by quantum tunneling, it grows rapidly.
The nucleation probability is maximized
when the direction of the soliton is perpendicular to
the external field.
We also find that decay (nucleation) is prohibited for
$B>B_c$ ($B<B_c$).
We find that
 the nucleation probabilities
calculated in the thin-defect approximation
and in the direct numerical simulations show
a good agreement in $2+1$ dimensions
once we read the vortex tension from the numerics.
On the other hand,  in $3+1$ dimensions, we find a difference between them at short distances at the subleading order
which we interpret as a remnant energy.

This paper is organized as follows.
In Sec.~\ref{sec:decay_wall}, we give a brief review
of quantum decay of a soliton in the complex $\phi^4$ model (the axion model with the domain wall number one)
without a topological term.
In Sec.~\ref{sec:nucleation_wall},
we present our model (the complex $\phi^4$ model with a topological term)
and discuss
quantum nucleation and decay probabilities of solitons
in the thin-defect approximation.
In Sec.~\ref{sec:numerical}
we numerically calculate the creation probabilities of solitons
in $2+1$ and $3+1$ dimensions
and compare those in the thin-defect approximation.
Section \ref{sec:summary} is devoted to a summary and discussion.
In Appendix \ref{sec:app2}, we present
an asymptotic behavior of the scalar field outside
a pair of vortex and an anti-vortex connected by a soliton.
In Appendix \ref{sec:app1}, we give a derivation of some formula
used in the quantum nucleation.

\section{Quantum decay of solitons by nucleation of holes: a review}
\label{sec:decay_wall}

We start with giving a brief review of the quantum decay of
solitons (domain walls)
 by quantum nucleations of holes
in a complex $\phi^4$ model (an axion model with the domain  wall number one).
The minimal model in $3+1$ dimensions is
\begin{eqnarray}
{\cal L}_{\rm UV} = \left|\partial_\mu \phi\right|^2 - \frac{\lambda}{4}\left(|\phi|^2 - v^2\right)^2 + v m^2(\phi + \phi^*),
\label{eq:Lag_axion}
\end{eqnarray}
with $v$ and $m$ are parameters whose mass dimension is 1, and $\lambda$ is dimensionless.
If the third term in Eq.~(\ref{eq:Lag_axion}) is absent, the model is
the Goldstone model invariant under a global $U(1)$ transformation $\phi \to e^{i\eta}\phi$
spontaneously broken in the homogeneous vacuum $|\phi| = v$.
There is a Nambu-Goldstone (NG) mode and
a Higgs mode whose mass is $m_h = v\sqrt{\lambda}$.
When we turn on the third term, the $U(1)$ symmetry is explicitly broken,
leaving the unique vacuum
where the NG mode becomes a pseudo-NG mode with the mass $m$.
The $U(1)$ symmetry is
an approximate symmetry when the mass of the pseudo-NG mode is sufficiently small
\begin{eqnarray}
m_h \gg m
\quad \Leftrightarrow \quad m^2 \ll \lambda v^2.
\label{eq:approxU(1)_cond}
\end{eqnarray}
There the vacuum expectation value can be approximated as
$
\left<\phi\right>  \simeq v \left( 1+ \frac{m^2}{m_h^2}\right).
$

The model admits two kinds of solitonic objects,
namely vortices and solitons (or domain walls).
The vortex is a global string with co-dimension two which is a topological defect if the explicit $U(1)$ breaking term is absent.
Thickness of the string and the tension of the string for $m=0$, namely the energy per unit length, are given by
\begin{eqnarray}
\delta_{\rm st} \sim m_h^{-1},\qquad
\mu\big|_{m\to 0} \sim \pi v^2\log (m_h L),
\label{eq:global_string}
\end{eqnarray}
where $L$ is a long distance cutoff.
When the $U(1)$ breaking term is not zero, the string is no longer topological and it is
always accompanied by the soliton which is a wall-like object with co-dimension one.

Probably the soliton can be most clearly seen in the limit of $\lambda \to \infty$
where the wine bottle potential becomes infinitely steep ($m_h \to \infty$), so that the amplitude of $\phi$ freezes out as $|\phi|=v$.
Writing $\phi = v e^{i\theta}$ and plugging it
into Eq.~(\ref{eq:Lag_axion}), we are lead to the sine-Gordon model
\begin{eqnarray}
{\cal L}_{\rm IR} = v^2 \left[(\partial_\mu\theta)^2 + 2 m^2 (\cos \theta -1)\right],
\label{eq:Lag_sG0}
\end{eqnarray}
where we have subtracted the constant $2m^2v^2$ for the minimum of the potential energy to be 0 for convenience.
The ground state is homogeneous as $\theta = 0$ with the redundancy of $2\pi n$  $(n \in \mathbb{Z})$.
The ground state energy $E^{({\rm vac})}=0$.
In addition, there is a sine-Gordon soliton
which we take perpendicular to the $z$-axis
without loss of generality:
\begin{eqnarray}
\theta = 4 \tan^{-1}e^{m z}.
\end{eqnarray}
This connects $\theta = 0$ at $z\to-\infty$ and $\theta = 2\pi$ at $z\to\infty$.
The thickness and the tension, namely the energy per unit area, of the soliton are given by
\begin{eqnarray}
\delta_{\rm dw} = m^{-1},\qquad
\sigma\big|_{\lambda \to \infty} = 16 m v^2.
\label{eq:sG_tension}
\end{eqnarray}
Note that the soliton in the sine-Gordon limit is classically stable but
it could be quantum-mechanically unstable.
This is because
it can end on
a sting which is infinitely thin
($\delta_{\rm st} \to 0$) in the $\lambda \to \infty$ limit, so that
holes surrounded by the strings can be created by quantum tunneling effect.

The instability of the soliton for a finite $\lambda$ is
two fold: 1) classical instability and 2) quantum instability.

1) The soliton in the UV theory is a metastable non-topological soliton. This is because a loop surrounding the $S^1$ vacuum manifold
which slightly slants by $vm^2(\phi + \phi^*) \sim 2 m^2v^2 \cos\theta$
can pass slip the potential barrier around $|\phi|=0$ and shrink to the unique vacuum.
This is the classical instability of the soliton
in the UV theory with finite $\lambda$. Roughly speaking,
if the approximate $U(1)$ condition in Eq.~(\ref{eq:approxU(1)_cond}) is satisfied,
the soliton remains classically meta-stable.
As for the tension of the string,
it is quite different for the massive case $m\neq 0$
from the massless case $m=0$ in which the tension is logarithmically
divergent as in Eq.~(\ref{eq:global_string}).\footnote{
In Ref.~\cite{Preskill:1992ck},
a string is attached by
a sine-Gordon domain soliton as our case,
but it is assumed that the energy of the vortex
is logarithmically divergent.}
The key point is that
the amplitude of the scalar field
 converges to the VEV exponentially fast
 as numerically confirmed in Appendix \ref{sec:app2},
 in contrast to the massless case for which
 the amplitude polynomially approaches to the VEV.
Thus,
the tension is finite
\begin{eqnarray}
\mu\big|_{m > 0} = \text{const.} \label{eq:mu}
\end{eqnarray}
 in contrast to the massless case in Eq.~(\ref{eq:global_string}).
This fact is crucial for the nucleation of topological soliton.
Note that the approximate $U(1)$ condition in Eq.~(\ref{eq:approxU(1)_cond}) implies $\delta_{\rm st} \ll \delta_{\rm dw}$.

2) Even when the soliton is classically metastable, it would quantum mechanically be unstable
because of the nucleation of a hole.
Let us assume a shape of a hole is circular.
If the radius $R$ of the hole is much greater than the soliton thickness $\delta_{\rm dw}$,
we can use the thin-defect approximation, providing
the decay probability \cite{Preskill:1992ck}
\begin{eqnarray}
P_\text{decay} = A e^{-S},\quad S = \frac{16 \pi \mu^3}{3\sigma^2},\quad R = \frac{2\mu}{\sigma}
\end{eqnarray}
where $\mu$ is the constant tension of the string and
the soliton tension $\sigma$ is well approximated by
Eq.~(\ref{eq:sG_tension}) if $\lambda \gg 1$.
The thin-defect approximation is justified for $R\sim \mu/\sigma \sim \mu/(mv^2) \gg 1/m$,
namely $\mu/v^2 \gg 1$.
Thus, the value of the bounce action $S$
is justified in the thin defect approximation if
$S \sim
\mu^3/\sigma^2 \sim \mu^3/(m^2v^4) = (v^2/m^2)(\mu/v^2)^3 \gg v^2/m^2$.
On the other hand, the classical stability requires $v^2/m^2 \gg 1/\lambda$.
Hence,
since the bounce action can be of order one or larger
depending on the parameters,
the
decay rate can be either large or smaller, respectively.

\section{Quantum nucleation and decay of solitons in external fields
in the thin-defect approximation
}\label{sec:nucleation_wall}

In this section, we give the models
(sine-Gordon model and complex $\phi^4$ model in the external field) in Subsec.~\ref{sec:model}
and estimate
nucleation probability of solitons
in any dimensions $d$
in terms of
tensions of solitons and strings (vortices)
in the thin-defect approximation in Subsec.~\ref{sec:thin-defect-approx}.
We also calculate decay probability of solitons in the external field
in Subsec.~\ref{sec:decay}.

\subsection{
The models
with external fields}\label{sec:model}
We consider the sine-Gordon model under a constant background field ${\bm B}$ in $3+1$ dimensions, given by
\begin{eqnarray}
{\cal L}_{\rm IR} = v^2 \left[(\partial_\mu\theta)^2 + 2 m^2 (\cos \theta -1) + c {\bm B} \cdot \nabla \theta\right],
\label{eq:Lag_sG}
\end{eqnarray}
where the overall constant $v$ is dimensionful, $m$ is a mass parameter, and the mass dimension of $c$ is $-1$.
The last term is a total derivative and is a topological term.
This Lagrangian is a simplification of low-energy effective Lagrangians for various interesting systems: chiral magnets with
easy-axis anisotropy in which the topological term is the DM term,
or chiral Lagrangian for pions under
a strong magnetic field (rapid rotation)
in which the last term originates from
the CME (CVE).
Here, we use a notation of ${\bm B}$ for either a magnetic field, a rotation or DM term.
The Hamiltonian reads
\begin{eqnarray}
{\cal H}_{\rm IR} = v^2 \left[\dot\theta^2 + (\nabla \theta)^2 - 2 m^2 (\cos \theta - 1) - c {\bm B} \cdot \nabla \theta\right].
\end{eqnarray}

Since the last term in Eq.~(\ref{eq:Lag_sG}) is of the first order in the derivative, it does not affect the equation of motion (EOM).
Indeed, the homogeneous configuration $\theta = 2\pi n$ remains the solution of EOM. The energy density is also unchanged from zero.
The soliton solution also remains  to be the same
\begin{eqnarray}
\theta = 4 \tan^{-1}e^{m \hat {\bm n}\cdot \bm{x}},
\end{eqnarray}
where we have introduced an arbitrary unit vector $\hat{\bm n}$
perpendicular to the soliton.
The solution itself is unchanged, however,
the soliton tension receives a correction from the background field.
Let $\alpha$ be relative angle between $\hat{\bm n}$ and the constant background field ${\bm B}$. Then, the additional energy
per unit area of the single soliton reads
\begin{eqnarray}
\delta \sigma
= - 2\pi v^2 c B \cos \alpha,
\end{eqnarray}
where $B = |{\bm B}|$, and we have used the fact that $\theta$ increases by $2\pi$ when one traverses the soliton
along $\hat{\bm n}$.
The net tension of the single soliton reads
\begin{eqnarray}
\sigma = 16 mv^2 - 2\pi v^2 c B \cos \alpha.
\label{eq:sigma}
\end{eqnarray}
This is minimized when $\alpha = 0$ ($\pi$) for $c > 0$ ($<0$). Namely, the most stable soliton is perpendicular to the
external field ${\bm B}$.
This implies that the soliton is tensionless at the critical value
\begin{eqnarray}
B = \frac{8m}{\pi |c|} \equiv B_{\rm c}.
\end{eqnarray}
Moreover, it is negative for $B > B_{\rm c}$.
Therefore, the homogeneous configuration $\theta = 2\pi n$ is no longer ground state, but
the soliton is the true ground state when $B > B_{\rm c}$.
The multiple solitons are created by increasing $B$, and in general the ground state is a periodic lattice of soliton which
is called the CSL.

The CSLs have been recently studied in various fields. However, most of the previous arguments are static and they do not address
how the homogeneous ground state is replaced by the soliton when $B$ increases from the value below $B_{\rm c}$ to the one above $B_{\rm c}$. Is the infinitely large soliton suddenly created at the moment of $B=B_{\rm c}$?
This sounds quite unphysical. In order to answer to this elementary question, we study quantum nucleation of the soliton
in this paper.

However, the sine-Gordon model in Eq.~(\ref{eq:Lag_sG}) is not the most suitable for that purpose.
This is because the soliton is topologically stable within the framework of the sine-Gordon model,
and so we cannot discuss neither decay nor nucleation.
Thus, we are naturally guided to a linear sigma model as a UV completion\footnote{From the viewpoint of the microscopic fundamental
theory like QCD, the Lagrangian (\ref{eq:Lag_UV}) should be regarded as a low energy effective theory.}
by including a massive degree
of freedom (the Higgs mode).
We consider a complex $\phi^4$ model (the axion model
with the domain wall number one)
with a constant background field ${\bm B}$:
\begin{eqnarray}
{\cal L}_{\rm UV} = \left|\partial_\mu \phi\right|^2 - \frac{\lambda}{4} \left(|\phi|^2 - v^2\right)^2 + v m^2(\phi + \phi^*) + c {\bm j}\cdot {\bm B}
\label{eq:Lag_UV}
\end{eqnarray}
with
\begin{eqnarray}
j^\mu = -\frac{i}{2}(\phi^*\partial^\mu\phi - \phi\partial^\mu\phi^*)  = |\phi|^2 \partial^\mu\theta\,,\qquad
\phi = |\phi|e^{i\theta}.
\end{eqnarray}
This reduces back to the sine-Gordon model in Eq.~(\ref{eq:Lag_sG}) in the limit $\lambda \to \infty$.
Since the last term of Eq.~(\ref{eq:Lag_UV}) does not contribute to EOM, all the solutions for $c=0$ remain to be solutions for $c\neq0$.
Therefore, meta-stable solitons attached by strings should exist in the model (\ref{eq:Lag_UV}) with $c\neq 0$ if
the approximate $U(1)$ condition in Eq.~(\ref{eq:approxU(1)_cond}) is satisfied.

\subsection{Nucleation probability of a soliton
in the thin-defect approximation}\label{sec:thin-defect-approx}
Now let us take $B$ which is ssmaller than $B_c$. The ground state should be the homogeneous configuration.
Then, we increase $B$ to any value above $B_c$ instantaneously.
The ground state should be solitonic in this case.
To estimate the probability of nucleation of a soliton, we reverse the arguments about the soliton decay in Sec.~\ref{sec:decay_wall}
in which the nucleation probability of {\it a hole on the domain wall}
was calculated, see Fig.~\ref{fig:decay_vs_nucleation}.
\begin{figure}[th]
\begin{center}
\includegraphics[width=10cm]{decay_vs_nucleation.pdf}
\caption{The schematic picture of the decay of soliton by nucleating a hole (left) and
the creation of soliton by nucleating a disk soliton (right).
Grey objects stands for the solitons and
the red objects corresponds to the strings.}
\label{fig:decay_vs_nucleation}
\end{center}
\end{figure}

Here, we consider {\it a disk of a soliton bounded by a string loop
in the homogeneous vacuum}.
Let us consider the spatial dimension $d$,
and later we will set $d=2,3$.
In the thin-defect approximation,
the bounce action  reads
\begin{eqnarray}
	S(R) = {\rm vol}(S^{d-1}) R^{d-1} \mu + {\rm vol}(B^{d}) R^d \sigma,
\label{eq:bounce_action}
\end{eqnarray}
where $R$ is the radius of the soliton,
and
${\rm vol}(S^{d-1}) R^{d-1}$ and  ${\rm vol}(B^{d})$
are volumes of the unit hypersphere and hyperball, given by
\begin{eqnarray}
{\rm vol}(S^{d-1}) = \frac{2 \pi ^{\frac{d}{2}}}{\Gamma \left(\frac{d}{2}\right)},\qquad
{\rm vol}(B^{d}) = \frac{\pi ^{d/2}}{\Gamma \left(\frac{d+2}{2}\right)},
\end{eqnarray}
respectively.
Note that the string tension $\mu$ is not logarithmically divergent but a finite constant
in the presence of the $U(1)$ breaking term (the third term in Eq.~(\ref{eq:Lag_UV})).
See Appendix \ref{sec:app2} for some details.

Clearly, $\mu$ is always positive.
In contrast, the soliton tension $\sigma$ given in Eq.~(\ref{eq:sigma}) can be either positive or negative.
In the absence of the topological term ($c=0$),
$\sigma$ is positive,
and $S$ has no stationary points except for $R=0$.
Namely, quantum nucleation of the disk is prohibited.

However, the situation drastically changes in the presence of the topological term, $c \neq 0$, because the soliton tension
can be negative $\sigma < 0$ for $B > B_{\rm c}$. Then, a nontrivial stationary point exists at
\begin{eqnarray}
R = \frac{d-1}{d}\frac{{\rm vol}(S^{d-1})}{{\rm vol}(B^{d})}\frac{\mu}{(-\sigma)}.
\end{eqnarray}
and the nucleation probability can be calculated as
\begin{eqnarray}
P_{\text{nucleation}} = A e^{-S},\quad
S = \frac{(d-1)^{d-1}}{d^d}\frac{\left[{\rm vol}(S^{d-1})\right]^d}{\left[{\rm vol}(B^{d})\right]^{d-1}}\frac{\mu^d}{(-\sigma)^{d-1}}.
\label{eq:S0}
\end{eqnarray}
Since
$-\sigma$
given in Eq.~(\ref{eq:sigma})
is maximized at $\alpha = 0$ ($\pi$) for $c > 0$ ($<0$), the bounce action is minimized
there with the negative soliton tension
\begin{eqnarray}
\sigma_\perp
= 16 mv^2 \left(1 -\frac{B}{B_{\rm c}}\right) < 0.
\end{eqnarray}
Therefore, the nucleation probability is maximized for the soliton
perpendicular to ${\bm B}$. Once the disk perpendicular to ${\bm B}$ is nucleated, it rapidly expands.
The thin-defect approximation is justified for $R \gg \delta_{\rm dw}$.
This can be rewritten as $\mu/(-\sigma_\perp) \gg 1/m$.

\subsection{Decay probability of a soliton in external fields
in the thin-defect approximation}
\label{sec:decay}

Here we consider quantum decay of a soliton in the external field.
Consider an infinitely large flat soliton perpendicular to the external field
under the external field larger than $B_{\rm c}$.
The bounce action of a hole on a soliton
can be written
in the thin-defect limit as
\begin{eqnarray}
S(R) = {\rm vol}(S^{d-1}) R^{d-1} \mu - {\rm vol}(B^{d}) R^d \sigma_0\left(1 - \frac{B}{B_{\rm c}}\right),
\end{eqnarray}
with $\sigma_0 = 16mv^2$. Since the second term is positive for $B> B_{\rm c}$,
the bounce action does not have stationary point, and therefore the decay
is {\it prohibited}.
We should emphasized that
the soliton which is metastable for $B=0$ is
completely stable for $B> B_{\rm c}$.

On the other hand, when we instantaneously decrease $B$ below $B_{\rm c}$, the stationary point of the bounce action
appears at
\begin{eqnarray}
R = \frac{B_{\rm c}}{B_{\rm c}-B} \frac{d-1}{d}\frac{{\rm vol}(S^{d-1})}{{\rm vol}(B^{d})}\frac{\mu}{\sigma_0},
\end{eqnarray}
and the value of the action reads
\begin{eqnarray}
S =
\left(\frac{B_{\rm c}}{B_{\rm c}-B}\right)^{d-1}
\frac{(d-1)^{d-1}}{d^d}\frac{\left[{\rm vol}(S^{d-1})\right]^d}{\left[{\rm vol}(B^{d})\right]^{d-1}}\frac{\mu^d}{\sigma_0^{d-1}}.
\end{eqnarray}
Comparing this with the bounce action without external field, we find
\begin{eqnarray}
S\big|_{B=0} < S\big|_{0<B<B_{\rm c}}.
\end{eqnarray}
This implies that the decay rate of the soliton is suppressed by the external field.
As the external field $B$ increases toward $B_{\rm c}$ from below, the action diverges and the quantum decay is strongly suppressed.

\section{Numerical simulations for quantum nucleation of solitons} \label{sec:numerical}
In this section, we numerically calculate nucleation probability
of solitons.
In Subsec.~\ref{eq:preliminary} we rewrite the Lagrangian
in terms of dimensionless variables.
In Subsecs.~\ref{sec:nucleation2+1} and \ref{sec:nucleation3+1},
we calculate nucleation probabilities of solitons
by numerically constructing bounce solutions
in 2+1 and 3+1 dimensions, respectively.

\subsection{Preliminary}\label{eq:preliminary}

A great benefit of considering the linear sigma model in Eq.~(\ref{eq:Lag_UV}) is that we can treat the soliton and strings
as  regular objects of finite sizes. With them at hand we can go beyond the thin-defect limit.

We will numerically solve EOM of Lagrangian in Eq.~(\ref{eq:Lag_UV}).
To this end, it is useful to rewrite Eq.~(\ref{eq:Lag_UV}) in terms of the dimensionless variables
\begin{eqnarray}
\tilde x^\mu = m x, \quad \tilde \phi = v^{-1} \phi,\quad \tilde \lambda = \frac{m_h^2}{m^2},\quad
\tilde {\bm B} = m^{-1} c{\bm B}.
\end{eqnarray}
Then, we have the Lagrangian
\begin{eqnarray}
{\cal L}_{\rm UV}
= m^2 v^2 \left[
|\tilde\partial_\mu \tilde\phi|^2 - \frac{\tilde\lambda}{4} \left(|\tilde\phi|^2 - 1\right)^2 + \tilde\phi + \tilde\phi^* + \tilde {\bm j}\cdot \tilde{\bm B}
\right]
\end{eqnarray}
where
$\tilde \lambda$ is the unique parameter characterizing solutions. For the (meta-)stable solitons and strings to exist, we need
to assume $\tilde\lambda \gg 1$ corresponding to the condition in Eq.~(\ref{eq:approxU(1)_cond}).
For concreteness, we will assume that the soliton is perpendicular to the $z$-axis. Therefore, the last term
in the bracket can be written as
\begin{eqnarray}
\tilde {\bm j}\cdot \tilde{\bm B} = \tilde B \tilde j_z \cos \alpha.
\end{eqnarray}
The Hamiltonian reads
\begin{eqnarray}
{\cal H}_{\rm UV} = m^2 v^2 \tilde {\cal H}_{\rm UV},\quad
\tilde {\cal H}_{\rm UV} =
\left|\tilde\nabla \tilde \phi_z\right|^2
+ \frac{\tilde\lambda}{4} \left(|\tilde\phi|^2 - 1\right)^2
- (\tilde\phi + \tilde\phi^*) -  \tilde j_z \tilde B \cos \alpha.
\label{eq:dimless_H}
\end{eqnarray}

\subsection{Quantum nucleation of a soliton in $2+1$ dimensions}
\label{sec:nucleation2+1}

Here, we investigate nucleation of solitons in $2+1$ dimensions,
in which the soliton is a linear object and the vortex is a particle object.
The $2+1$ dimensional version of Eq.~(\ref{eq:bounce_action}) with $d=2$ is
\begin{eqnarray}
S = 2\pi R \mu + \pi R^2 \sigma.
\label{eq:S_2+1}
\end{eqnarray}
Its extremum is given by Eq.~(\ref{eq:S0}) for $d=2$,
\begin{eqnarray}
R_0 = \frac{\mu}{-\sigma},\qquad S_0 = \frac{\pi \mu^2}{-\sigma}.
\label{eq:R0S0_2d}
\end{eqnarray}
Note that we have $[v]=\frac{1}{2}$, $[\lambda] = 1$, and $[m]=1$ in $2+1$ dimensions.
We will compare this analytic formula in the thin-defect limit with numerical simulations for the soliton with finite thickness.

Once we obtain a numerical solution for a soliton attached by a vortex and an anti-vortex at its both ends,
we can measure the dimensionless radius $\tilde R$ (a half length) of the soliton and evaluate
the dimensionless total energy $\tilde{\cal E}$ by
\begin{eqnarray}
\tilde {\cal E}(\tilde R) = \int d^2\tilde x\  \tilde {\cal H}_{\rm UV}(\tilde R).
\end{eqnarray}
From this, we can evaluate the bounce action through the following formula
\begin{eqnarray}
S
= \int d^3x\, {\cal H}_{\rm UV}
= \frac{v^2}{m} \int d^3\tilde x\, \tilde{\cal H}_{\rm UV}
=
\alpha_1 \frac{v^2}{m} \int_0^{\tilde R} d\tilde r~\tilde{\cal E}(\tilde r),
\label{eq:EtoSd=2}
\end{eqnarray}
with a constant $\alpha_1 = \pi$ (see Appendix \ref{sec:app1} for a derivation).
To understand the formula quickly, let us substitute the energy formula $\tilde{\cal E}(\tilde R) = 2\tilde\mu + 2\tilde R \tilde \sigma$
with $\tilde \mu = \mu/v^2$ and $\tilde \sigma = \sigma/(mv^2)$
in the thin-defect limit
(the soliton of the length $2\tilde R$ with two vortices). We easily find that
the bounce action in Eq.~(\ref{eq:S_2+1}) is correctly reproduced.

Note that the formula in Eq.~(\ref{eq:EtoSd=2}) is valid only for constant $\mu$. If $\mu$ depended on $R$ logarithmically as
the usual global vortex, we cannot use Eq.~(\ref{eq:EtoSd=2}).
In Appendix \ref{sec:app2}, we show our numerical solution
in which the profile of the scalar field
exponentially converges to the VEV in the asymptotic region,
in contrast to the usual global vortex without any domain walls
for which the profiles polynomially approaches to the VEV.

By differentiating $S$ by $R$, we have
\begin{eqnarray}
\frac{dS}{dR}
= \alpha_1 \frac{v^2}{m}\frac{d\tilde R}{dR} \tilde{\cal E}(\tilde R)
= \alpha_1 v^2 \tilde{\cal E}(\tilde R).
\end{eqnarray}
The extremum of $S$ is then identified with the zero of ${\cal E}$:
\begin{eqnarray}
\frac{dS}{dR} = 0
\quad\Leftrightarrow\quad
\tilde {\cal E}(\tilde R) = 0.
\end{eqnarray}

The remaining task is constructing suitable numerical configurations with a soliton bounded by a vortex and an anti-vortex.
To this end, we use the standard relaxation scheme. Our method consists of two steps. Firstly, we take a product
ansatz of a pair of a vortex and an anti-vortex separated at distance $2\tilde R_{\rm ini}$ as an initial configuration of the relaxation.
At this stage,
we fix the positions of the vortices. Then, the straight soliton of the length $2\tilde R_{\rm ini}$ is generated and the configuration converges quite soon.
We use this convergent configuration as the initial configuration for the second relaxation,
in which
we do not fix the vortex positions.
During the second relaxation process, the vortices
approach to each other due to
the soliton tension, and eventually annihilate each other.
We repeatedly measure the distance $2\tilde R$ of the vortices and compute $\tilde{\cal E}(\tilde R)$.

\begin{figure}[t]
\begin{center}
\includegraphics[width=15cm]{amplitude_2dim.pdf}
\caption{The soliton attached by a vortex and anti-vortex in $2+1$ dimensions. The negative amplitude $(-|\tilde\phi|)$ is plotted.}
\label{fig:amplitude_2dim}
\end{center}
\end{figure}
To be concrete,
we take $\tilde \lambda = 100$ which is large enough for the soliton and vortices to be classically metastable.
The amplitudes $(-|\tilde\phi|)$ for several different separations are shown in Fig.~\ref{fig:amplitude_2dim}.
The two peaks correspond to the vortex and anti-vortex while the linear object stretching between them is the soliton
which is visible only for a large $\tilde R$.
We evaluate $\tilde{\cal E}$ with three different values of $\tilde B \cos\alpha = \{1,1/2,1/4\}$ in Eq.~(\ref{eq:dimless_H}).

\begin{figure}[th]
\begin{center}
\includegraphics[width=15cm]{EvsR_2dim.pdf}
\caption{The dimensionless total energy $\tilde{\cal E}$ is plotted as the function of the dimensionless radius $\tilde R$.
Three different choices of $\tilde B \cos\alpha = \{1,1/2,1/4\}$
corresponding to $B \geq B_c$, $B = B_c$ and $B \leq B_c$, respectively, are shown. The right panel is the zoom-up around $\tilde R=0$ of the left one.}
\label{fig:EvsR_2dim}
\end{center}
\end{figure}
The results are shown in Fig.~\ref{fig:EvsR_2dim}. The datas are well fitted by a linear function in the large $\tilde R$ region
\begin{eqnarray}
\tilde{\cal E} = 2\tilde R a  + 2 b,
\label{eq:fit_2dim}
\end{eqnarray}
which should be compared with the thin-defect limit $\tilde{\cal E} = 2\tilde R \tilde\sigma  + 2 \tilde\mu$.
The coefficient $a$ can be either positive or negative because it is related to the soliton tension
which depends on $\tilde B \cos\alpha$.
On the other hand, the constant $b$ should be insensitive on $\tilde B \cos \alpha$ because it should be identified with the vortex tension
which is independent on the background field.
Indeed, our numerical solutions show that the three lines almost meet at $\tilde R = 0$ in Fig.~\ref{fig:EvsR_2dim}.
The coefficients read numerically are shown in Table.\ref{tab:ab_2dim}.
Thus we numerically determine the tensions $\tilde\mu$ and $\tilde\sigma$. Importantly, $\tilde\mu$ is a constant as we mentioned above,
see also Appendix \ref{sec:app2}.
\begin{table}
\begin{center}
\begin{tabular}{c|cc}
$\tilde B \cos\alpha~(d=2)$ & $a(\sim \tilde \sigma)$ & $b(\sim \tilde \mu)$ \\
\hline
$1$ & $- 15.6$ & $11.6$\\
$1/2$ & $0.07$ & $12.0$\\
$1/4$ & $7.85$ & $12.2$
\end{tabular}
\qquad
\begin{tabular}{c|ccc}
$\tilde B \cos\alpha~(d=3)$ & $a(\sim \tilde \sigma)$ & $b(\sim \tilde \mu)$ & $c$ \\
\hline
$1$   & $-15.3$ & $11.8$ & $45.6$\\
$1/2$ & $-0.15$ & $12.7$ & $28.4$\\
$1/4$ & $7.78$ & $12.3$ & $24.3$
\end{tabular}
\caption{The numerically obtained coefficients in Eqs.~(\ref{eq:fit_2dim}) and (\ref{eq:fit_3d}).}
\label{tab:ab_2dim}
\end{center}
\end{table}

Among the three different choices, only $\tilde B \cos \alpha = 1$ leads to the negative soliton tension,
corresponding to the case of $B \geq B_c$.
The stationary point is $\tilde R_0 \simeq 0.75$, and the value of the bounce action is about $\tilde S \simeq b\tilde R \simeq 9.0$.
Hence, the nucleation probability
can be estimated as
\begin{eqnarray}
P_{\rm nucleation} = A \exp\left(-\alpha_1\frac{v^2}{m} \times 9.0\right).
\end{eqnarray}
Note that the numerically determined values $(\tilde R_0,\tilde \sigma, \tilde\mu) = (0.75, - 15.6, 11.6)$ is
consistent with the analytic formula $\tilde R_0 = \tilde \mu/(-\tilde\sigma)$ for the thin-defect limit.

\subsection{Quantum nucleation of a soliton in $3+1$ dimensions}
\label{sec:nucleation3+1}

Next, we numerically investigate quantum nucleation of a disk soliton bounded by a string loop in $3+1$ dimensions.
By putting $d=3$ in Eq.~(\ref{eq:bounce_action}), the bounce action of the thin-defect limit is given by
\begin{eqnarray}
S = \pi R^2 \mu + \frac{4\pi}{3} R^3 \sigma,
\label{eq:S_3+1}
\end{eqnarray}
and its extremum is given by Eq.~(\ref{eq:S0}) for $d=3$,
\begin{eqnarray}
 R_0 = \frac{2\mu}{-\sigma},\qquad
 S_0 = \frac{16\pi \mu^3}{3\sigma^2}.
\label{eq:R0S0_d3}
\end{eqnarray}

The numerical procedures we adopt in this subsection are
the same as those in the previous subsection
except for differences due to the spacial dimensions.
We numerically evaluate
the dimensionless mass $\tilde{\cal E}$ for a disk of a soliton of the radius $R$
\begin{eqnarray}
\tilde{\cal E}(\tilde R) = \int d^3\tilde x~\tilde{\cal H}_{\rm UV},
\end{eqnarray}
where the dimensionless Hamiltonian $\tilde{\cal H}_{\rm UV}$ is given in Eq.~(\ref{eq:dimless_H}).
Then, we evaluate the bounce action $S$ by the following formula
\begin{eqnarray}
S(R)
= \int d^{4}x\, {\cal H}_{\rm UV}
= \frac{v^2}{m^{2}} \int d^{4}\tilde x\, \tilde{\cal H}_{\rm UV}
=
\alpha_2 \frac{v^2}{m^2}  \int_0^{\tilde R} d\tilde r~\tilde{\cal E}(\tilde r).
\label{eq:S_d=3}
\end{eqnarray}
The constant factor is $\alpha_2 = 4$ whose derivation is given in Appendix \ref{sec:app1}.
Again, we should note that this formula is valid for the constant $\mu$.
As a quick check of the formula, one can reproduce Eq.~(\ref{eq:S_3+1}) by substituting
$\tilde{\cal E} = 2\pi\tilde R\tilde\mu + \pi \tilde R^2 \tilde \sigma$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=15cm]{disk_initial.pdf}
\caption{The three-dimensional disk soliton perpendicular to the $z$-axis generated from the two-dimensional
linear soliton. In the left panel the red part shows the region for $|\tilde\phi|<0.9$, and
the blurred-blue part corresponds to the region for ${\rm Re}[\tilde\phi] < 0.4$.
The middle and right panels show $-|\tilde\phi|$ on the $\tilde x = 0$ and $\tilde z = 0$ cross sections, respectively.}
\label{fig:disk_initial}
\end{center}
\end{figure}
As before we take relatively large value $\tilde \lambda = 100$ to assure the classical stability of the solitons
and strings. In order to prepare a disk shape soliton, we recycle the numerical configuration of
the linear soliton attached by two vortices in $2+1$ dimensions.
Set the two-dimensional linear soliton along the $x$-axis on the $xz$-plane, and let $\phi_{\rm 2d}(x,z)$ be the
corresponding field configuration. Then, a three-dimensional disk soliton perpendicular to the $z$-axis can be obtained by
rotating it around the $z$-axis, namely
$\phi_{\rm 3d}(x,y,z) = \phi_{\rm 2d}(x\cos\theta+y\sin\theta,z)$ with $\tan \theta =y/x$, see Fig.~\ref{fig:disk_initial}.

Having this as an initial configuration, we evolve it by a standard relaxation method.
The disk soliton shrinks as the relaxation process proceeds. We measure the radius and the mass,
so that we determine the function $\tilde{\cal E}(\tilde R)$.
Finally, we calculate the bounce action $S(R)$ by plugging it to the formula in Eq.~(\ref{eq:S_d=3}).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=15cm]{rings_3d.pdf}
\caption{The disk soliton bounded by a ring string.
The red part shows the region for $|\tilde\phi|<0.9$, and
the blurred-blue part corresponds to the region for ${\rm Re}[\tilde\phi] < 0.4$.}
\label{fig:rings_3d}
\end{center}
\end{figure}
The time evolution of the disk soliton under the relaxation process is shown in Fig.~\ref{fig:rings_3d}.
The disk is initially large, and we can clearly observe a circular closed string  (red part: $|\tilde\phi|<0.9$)
and a disk soliton (blurred-blue part: ${\rm Re}[\tilde\phi] < 0.4$).
We determine the radius $\tilde R$ of the ring by seeking the point where $|\tilde\phi| = 0$.
Note that the energy does not immediately vanishes when $\tilde R$ reaches zero. This is a finite width effect which
is missed in the thin-defect limit. Since the soliton and string are regular objects with finite sizes,
a remnant of energy still exists and it gradually decays and finally disappears.

\begin{figure}[t]
\begin{center}
\includegraphics[width=15cm]{EvsR_3dim.pdf}
\caption{The dimensionless total energy $\tilde{\cal E}$ is plotted as the function of the dimensionless radius $\tilde R$ for $d=3$.
We set $\tilde B \cos\alpha = \{1,1/2,1/4\}$.
The right panel is the zoom-up near $\tilde R=0$ of the left one.}
\label{fig:EvsR_3dim}
\end{center}
\end{figure}
The numerical results for $\tilde B \cos\alpha = \{1,1/2,1/4\}$ are shown in Fig.~\ref{fig:EvsR_3dim}.
The fact that the energy of $\tilde B \cos\alpha = 1$ is negative for large $\tilde R$ indicates that the soliton tension is negative.
The zero of $\tilde{\cal E}$ is found as $\tilde R \simeq 2$ which should correspond to the extremum point of the bounce action.
The energy of $\tilde B \cos\alpha = 1/2$ is approximately a linear function of $\tilde R$ implying the soliton tension vanishes.
That of $\tilde B \cos\alpha = 1/4$ grows faster than linear functions, implying that the soliton tension is positive.
These are well fitted by
\begin{eqnarray}
\tilde{\cal E} = \pi \tilde R^2 a + 2\pi \tilde R b + c.
\label{eq:fit_3d}
\end{eqnarray}
The values of these coefficients are shown in Table \ref{tab:ab_2dim}.
The values of $a$ ($\sim \tilde \sigma$) and $b$ ($\sim\tilde \mu$)
are consistent between the left ($d=2$) and right ($d=3$) tables in Table~\ref{tab:ab_2dim}.
The last term $c$ corresponds to the remnant energy at $\tilde R=0$. This is absent in the thin-defect limit. Hence,
the bounce action for the finite size soliton is slightly larger than the one
at the thin-defect limit.

By using the fit in Eq.~(\ref{eq:fit_3d}), we can evaluate the action for $\tilde B \cos\alpha = 1$ as
\begin{eqnarray}
S = \alpha_2 \frac{v^2}{m^2} \int_0^{\tilde R_0=2} d\tilde r~\tilde{\cal E}(\tilde r) \simeq  111 \alpha_2 \frac{v^2}{m^2}.
\end{eqnarray}
Thus, the nucleation probability can be obtained as
\begin{eqnarray}
P_{\rm nucleation} = A \exp\left(- 111 \alpha_2 \frac{v^2}{m^2}\right).
\label{eq:nucleation-rate}
\end{eqnarray}

Before closing this section, let us examine the relevance of
the remnant energy found above.
If we fit the numerical data by ignoring the remnant energy with forcing $c=0$, then we find $a=-16.7$ and $b=16.7$,
see the red-dashed curve in Fig.~\ref{fig:EvsR_3dim}.
The value of integration for the constrained fit is $\int_0^{\tilde R_0=2} d\tilde r~\tilde{\cal E} \simeq 69$,
so that the nucleation probability is slightly increased. Here,
$a=-16.7$ is still consistent with the one obtained in the $d=2$ case, whereas
$b=16.7$ shows a relatively large discrepancy from $b$ in the $d=2$ case.
Hence, the string tension $b$ is not correctly captured by the constrained fit. Thus, we conclude that
the remnant energy $c$
included in the unconstrained fit is not a sort of artifacts of the numerical simulation but it should be
considered as a real finite width effect.

In conclusion, we have succeeded in numerically evaluating the bounce action for the soliton bounded by the string
with finite thickness. The finite width effect has been found and it slightly reduces the nucleation probability
compared with the thin-defect limit.

\section{Summary and discussion}\label{sec:summary}

We have proposed quantum nucleation of topological solitons
through quantum tunneling,
as a novel mechanism for formation of topological solitons.
We have discussed chiral solitons in a complex $\phi^4$ model (an axion model) with a topological term,
which is a low-energy theory of
chiral magnets with an easy-plane anisotropy
and QCD at finite density under strong magnetic field or rapid rotation.
First, we have estimated the creation probability analytically
in terms of tensions of string (vortex) and soliton
in the thin-defect approximation in any dimension.
Second,
we  have performed numerical simulations
in $2+1$ and $3+1$ dimensions by
the relaxation (gradient flow) method,
and have obtained creation probabilities.
We have found a good agreement between the thin-defect approximation
and direct numerical simulation in $2+1$ dimensions
and have found, in $3+1$ dimensions,
a difference between them at short distances at the subleading order, which can be interpreted as the remnant energy.

We have considered the complex $\phi^4$ model as an UV theory
for the sine-Gordon model at IR limit appearing in various context.
Sine-Gordon solitons are almost insensitive to UV,
but different UV theories give different structures of strings
(vortices).
However, creation probabilities will be insensitive to such details.

In this study, we have estimated
the nucleation probability in the vacuum
where there are no solitons.
In the case with
the external field
above the critical value,
the true ground state is a CSL.
Formation of the CSL in the homogeneous vacuum should occur in the following process.
Let us turn on the external field $B > B_{\rm c}$ in
the homogeneous vacuum.
Initially, disk solitons of the critical radius $R_0$
in Eq.~(\ref{eq:R0S0_d3})
are nucleated all over with the creation rate
in Eq.~(\ref{eq:nucleation-rate}).
They rapidly expand
as in the right panel of Fig.~\ref{fig:decay_vs_nucleation},
growing up to infinitely large solitons.
These solitons repel each other and thus
adjust intersoliton distances to minimize the total energy,
thereby eventually forming into a CSL
as schematically shown in Fig.~\ref{fig:CSL}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=15cm]{CSL.pdf}
\caption{Nucleation of solitons and formation of a CSL.}
\label{fig:CSL}
\end{center}
\end{figure}
Of course, this is a rough sketch and we need more
detailed analysis of
dynamical process.
We also need to calculate
nucleation probabilities of solitons not in
the homogeneous vacuum but also in inhomogeneous soliton backgrounds.
For instance, once the CSL ground state is formed,
nucleation probabilities of solitons should be zero
in such a background.
If we instantaneously increase (decrease) the external field
in the CSL ground state,
the number density of solitons
 should be decreased (increased).
We thus need nucleation (decay) rates of
solitons in the CSL background,
which remain as a future work.

In this paper, we have considered the Abelian sine-Gordon model
for simplicity.
On the other hand,
it was found in Ref.~\cite{Eto:2021gyy}
that
in the case of two-flavor baryonic matter under rotation,
non-Abelian solitons with non-Abelian ${\mathbb C}P^{N-1}$ moduli
\cite{Nitta:2014rxa,Eto:2015uqa}
are also present
in the ground state of QCD in a certain parameter region.
In this case, a non-Abelian soliton is bounded by
a non-Abelian global string \cite{Nitta:2014rxa, Eto:2021nle}.
In such a case, the creation probability may depend on
the dimension $N$ of the moduli as numerical factor.

In $2+1$ dimensions,
the pseudo-NG mode is mapped to an electromagnetic field
 under a duality,
while vortices are mapped to charged particles.
With nonzero mass $m$, particles are confined
by electric fluxes.
In this duality,  the topological term will be mapped to
a constant electric field.
It remains as a future problem to study nucleation probabilities
in terms of the duality.
In $2+1$ dimensions, there is a BKT transition
at finite temperature.
It is interesting to discuss
 whether there is any conflict between
 quantum nucleation and the BKT transition.

Note added:
While this paper is being completed, we were informed that
the authors of Ref.~\cite{Nishimura} was preparing
a draft which may have some overlap with our work.

\section*{Acknowledgements}
This work is supported in part by JSPS Grant-in-Aid for Scientific Research (KAKENHI Grant No. JP22H01221).
The work of M. E. is supported in part by the JSPS Grant-in-Aid for Scientific Research KAKENHI Grant No. JP19K03839 and
the MEXT KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas ``Discrete Geometric Analysis for Materials Design''
No. JP17H06462 from the MEXT of Japan.

\begin{appendix}

\section{Asymptotic behavior of the scalar field
of a vortex attached by a soliton}
\label{sec:app2}

Here, we show
a numerical solution with some more details to demonstrate that
the amplitude of $\phi$ for a vortex attached by
a soliton decays exponentially fast,
in contrast to the slow-tail with a negative power of
isolated global vortices without solitons.
For this purpose, we use a configuration of
a vortex and an anti-vortex connected by a soliton
discussed in Sec.~\ref{sec:nucleation2+1}, see Fig.~\ref{fig:amplitude_2dim}.
When the vortex and anti-vortex are well separated compared
with the width of the soliton,
the configuration around the vortex
is close to that of a single vortex attached by a soliton.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=15cm]{asym.pdf}
\caption{The left panel shows $\tilde\phi(\tilde x,0)$ for the configuration shown in the top-left
panel of Fig.~\ref{fig:amplitude_2dim}, and the right panel shows asymptotic behavior of the same solution.}
\label{fig:asym}
\end{center}
\end{figure}
The left panel of Fig.~\ref{fig:asym} shows
a scalar field profile
$\tilde\phi(\tilde x,0)$ at a cross section $y=0$ of
the configuration shown in the top-left
panel of Fig.~\ref{fig:amplitude_2dim} with $\tilde R=4.83$.
One can see that apart from the vortex core,
the scalar field $\tilde \phi$ quickly converges
to a constant in the vacuum.
To confirm that the asymptotic behavior is an exponential tail, we show
$\log\delta\tilde\phi = \log (\tilde v - |\tilde\phi|)$
in the right panel of Fig.~\ref{fig:asym}. We numerically fit the asymptotic tail and the result is
 $\delta\tilde\phi(\tilde x,0) \propto e^{-1.8\tilde x}$.
 Thus, the amplitude converges exponentially fast to the VEV.

We can confirm the same behaviors
in any directions from the vortex center except for
the directions of the soliton within its width.
This fact implies
 that the vortex tension is finite as in Eq.~(\ref{eq:mu}),
 in contrast to an isolated global vortex
 whose tension is logarithmically divergent
 as in Eq.~(\ref{eq:global_string}).
 Physically,
the logarithmically divergent behavior (of an isolated global vortex) is replaced by
 the soliton tension.
 This point was missed in the literature
 \cite{Preskill:1992ck,Vilenkin:2000jqa}
 in which the vortex tension was assumed to be
 logarithmically divergent even when the vortex is attached by a soliton.
 This is crucial to evaluate
 bounce actions in the thin-defect limit
 and likewise the decay rates
 and nucleation probabilities
 of topological solitons.

\section{A derivation of Eqs.~(\ref{eq:EtoSd=2}) and (\ref{eq:S_d=3})}
\label{sec:app1}

In this appendix, we give a derivation of Eqs.~(\ref{eq:EtoSd=2}) and (\ref{eq:S_d=3}) for the spatial dimension $d=2,3$.
However, we will consider generic $d$ below.
Since the number of codimensions of strings(vortices) is 2 and that of  solitons is 1, a soliton is a $(d-1)$-dimensional ball
$B^{d-1}$ while the string wrapping the soliton is a $(d-2)$-dimensional sphere $S^{d-2}$.
The volumes of unit $d$-sphere and $d$-ball are given by
\begin{eqnarray}
{\rm vol}(S^{d-1}) = \frac{2 \pi ^{\frac{d}{2}}}{\Gamma \left(\frac{d}{2}\right)},\qquad
{\rm vol}(B^{d}) = \frac{\pi ^{d/2}}{\Gamma \left(\frac{d+2}{2}\right)}.
\end{eqnarray}
Therefore, the mass in the thin-defect limit reads
\begin{eqnarray}
\tilde {\cal E}(\tilde R)
= \int d^{d}\tilde x\, \tilde{\cal H}_{\rm UV}
= {\rm vol}(S^{d-2}) \tilde R^{d-2} \tilde \mu + {\rm vol}(B^{d-1})  \tilde R^{d-1} \tilde \sigma,
\end{eqnarray}
with the dimensionless tensions of string and soliton
\begin{eqnarray}
\tilde \mu = \mu/v^2,\quad
\tilde \sigma = \sigma/(mv^2),
\end{eqnarray}
respectively.
On the other hand, the string world-volume is $S^{d-1}$ and the soliton world-volume is $B^{d-1}$
for the bounce action, and we have
\begin{eqnarray}
\tilde S(\tilde R)  \equiv \int d^{d+1}\tilde x\, \tilde{\cal H}_{\rm UV}
={\rm vol}(S^{d-1}) \tilde R^{d-1}\tilde \mu + {\rm vol}(B^{d}) \tilde R^{d} \tilde \sigma.
\end{eqnarray}
Integrating $\tilde {\cal E}(\tilde R)$, we find
\begin{eqnarray}
\int^{\tilde R}_0 d\tilde r~\tilde {\cal E}(\tilde r)
&=& \frac{1}{d-1}{\rm vol}(S^{d-2}) \tilde R^{d-1} \mu + \frac{1}{d}{\rm vol}(B^{d-1})  \tilde R^{d} \sigma \nonumber\\
&=& \left(\frac{1}{d-1}\frac{{\rm vol}(S^{d-2})}{{\rm vol}(S^{d-1})}\right) {\rm vol}(S^{d-1})\tilde R^{d-1} \mu
+ \left(\frac{1}{d}\frac{{\rm vol}(B^{d-1})}{{\rm vol}(B^{d})}\right) {\rm vol}(B^{d}) \tilde R^{d} \sigma. \;\;\;
\end{eqnarray}
Note that the two ratios are identical as
\begin{eqnarray}
\alpha_{d-1} \equiv (d-1) \frac{{\rm vol}(S^{d-1})}{{\rm vol}(S^{d-2})}
= d \frac{{\rm vol}(B^{d})}{{\rm vol}(B^{d-1})},
\end{eqnarray}
and therefore we have
\begin{eqnarray}
\tilde S(\tilde R) = \alpha_{d-1} \int^{\tilde R}_0 d\tilde r~\tilde {\cal E}(\tilde r).
\end{eqnarray}
We assume that this formula is valid not only for the thin-defect limit but also the case that the defects have regular sizes.
We have used $\alpha_1 = \pi$ and $\alpha_2 = 4$ in the text.

\end{appendix}

\bibliographystyle{jhep}

redpajama_set_name: RedPajamaArXiv


Listed below are international publications and chapters from the Minerals Yearbook Volume III: Area Reports�International�Asia and the Pacific. These annual reviews are designed to provide timely statistical data on mineral commodities in various countries. Each report includes sections on government policies and programs, environmental issues, trade and production data, industry structure and ownership, commodity sector developments, infrastructure, and a summary outlook.


text:
NICKY BOOTH
The Stewards and staff of the British Boxing Board of Control are very saddened to hear of the passing of Former British and Commonwealth Bantamweight Champion Nicky Booth.
Our sincerest condolences are sent to his family at this time.
Mr. Terry O'Connor
British Boxing Board of Control Judge Mr. Terry O'Connor appeared before the Stewards of the Board to discuss his performance at the tournament held on 17th October 2020 at the East of England Showground Arena, Peterborough.
Allegations made on social media and the print media as to Mr. O'Connor using a mobile telephone or similar handheld device were considered.
Following extensive consideration of footage and Mr. O'Connor's evidence supplied, the Stewards are satisfied that Mr. O'Connor was not in possession of any telephone or handheld device whilst carrying out his duties as a Judge in the contest between Lewis Ritson and Miguel Vazquez and his scoring of said bout was not affected in any way.
In addition, Mr. O'Connor's scoring of the bout was considered and whilst judging a contest is subjective and following Mr. O'Connor's explanation of how he scored the bout, the Stewards are satisfied that Mr. O'Connor's final score reflected his own opinion of the contest.
Judge Terry O'Connor
JUDGE TERRY O'CONNOR
Mr. O'Connor will appear before the Stewards of the British Boxing Board of Control following his appointment to the boxing tournament at the East of England Showground Arena on Saturday, 17th October 2020.
Alan Minter
The Stewards and staff of the British Boxing Board of Control are very saddened to hear of the passing of Former World Middleweight Champion Alan Minter.
BOXER BILLY JOE SAUNDERS
Billy Joe Saunders appeared before Stewards of the Board on Wednesday, 22nd July 2020 due to recent comments made on Social Media.
Following consideration of Mr. Saunders explanation, the Stewards found Mr. Saunders guilty of misconduct and have fined him £15,000.00p to be donated to charities.
The suspension of his boxer's licence has been lifted.

redpajama_set_name: RedPajamaCommonCrawl


When a home suffers a water damage event, a mold infestation can quickly arise and spread throughout a home in 48-72 hours. Because mold can produce allergens and irritants, you will want a professional that has training and experience to properly resolve the mold infestation. If you suspect that your Woodland Hills home or business has a mold problem, SERVPRO of South Tulsa County can inspect, assess and remediate your property.
We live and work in this community too; we might even be neighbors. As a locally owned and operated business, SERVPRO of South Tulsa County is close by and ready to respond to your mold damage emergency. We are proud to be an active member of the Woodland Hills community and want to pitch in and help make our community the best it can be.
To keep the mold spores from spreading further, we will use various mold containment procedures. Our professionals may use physical barriers and negative air pressure to isolate the contaminated area and to help prevent the mold spores from spreading through your Woodland Hills, OK home or business. Also, we'll turn off the property's fans and heating and cooling systems to prevent the spread of mold.
SERVPRO of South Tulsa County will use a variety of cleaning techniques and products to clean your furnishings and personal belongings. Since mold infestations also cause a persistent musty odor, our professionals are also trained to remove odors and provide deodorization using fogging equipment.
SERVPRO of South Tulsa County has the expertise to make any mold problem a memory in Woodland Hills, OK. We have seen the distress homeowners experience upon finding mold at their property. SERVPRO can help calm any mold concerns. Our trained professionals have the equipment and knowledge it takes to properly remediate mold. Call (918) 437-2095 - We are ready and waiting to respond to your emergency.


Our foamboard dimensions here in switzerland are 50x70cm or 100x70cm and 140x100cm.
The original plans are oke, but we wasting a lot foamboard at the first self cutting plane.
I modified the plan for our Need (Pusher Version), my plan size is 100x70cm. So it fit perfect for 1 plane.
The foam spar holes are only a 50% score cut cause i pulled out the foam from the inner side, then we have not the 2 holes on the underside of the plane. I also mark the Balancepoints blue, we just mark them on the plane with a small dot or a neddle.


text:
Christopher Sheppard
Follow Christopher Shepp ...
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The Party Review
Comedies don't get much darker than this pitch-black British movie, written and directed by Sally Potter (Ginger & Rosa) with lacerating irony and such a furious sense of humour that it's impossible to stifle our laughter no matter how we try. Impeccably played by a great cast, it's a lot like watching a play, as it unfolds in real time in a single setting with just seven characters. But Potter's decision to film it in black and white adds a sharp edge of surrealism that makes it also feel like a classic.
It opens as Janet (Kristin Scott Thomas) is preparing for a small dinner party to celebrate her appointment as a government minister. With something else on his mind, her husband Bill (Timothy Spall) is completely drunk before the first guest arrives, but Janet doesn't really notice. Her outspoken best friend April (Patricia Clarkson) turns up first with her German philosopher boyfriend Gottfried (Bruno Ganz). Next is feminist professor Martha (Cherry Jones) and her younger girlfriend Jinny (Emily Mortimer), who is pregnant with triplets. And finally it's the banker Tom (Cillian Murphy), hopped up on cocaine with a gun in his pocket. His wife is running late. And over the next hour, everyone lets a few secrets out of the bag.
Continue reading: The Party Review
Ginger And Rosa Review
An extraordinary cast lifts this grim British drama into something watchable, even if the script ultimately gives up trying to make any sense. The main problem is that the story is very badly fragmented, but it still captures a vivid sense of how it felt to grow up in 1962 Britain. And the actors give performances that bring the characters to life even in scenes that are somewhat melodramatic.
Ginger and Rosa (Fanning and Englert) are inseparable 16-year-olds who were born in the same hospital on the same day. As they both ponder the horrific possibilities of the Cold War, their reactions begin to diverge, perhaps their first disagreement ever. Ginger's parents (Hendricks and Nivola) are liberal-minded and about to separate yet again, so she takes a militant approach to stopping nuclear annihilation. Rosa lives with her deeply religious single mother (May) and believes that the only thing to do is pray about it. But the thing that drives a real wedge between the girls is Ginger's suspicion that her dad might be having an affair with Rosa.
In the early scenes, Potter establishes the girls as imaginative friends with free spirits who do everything together. Then the plot begins to take increasingly dark twists and turns, leading to a series of awkward or downright horrible confrontations that are freaky and emotional but also thoroughly mawkish. There's a lot of glowering and weeping on display from everyone on-screen. Fortunately Fanning and newcomer Englert maintain a loose honesty in their performances that helps carry us through the difficult moments. And the starry supporting cast is terrific.
Continue reading: Ginger And Rosa Review
Yes Review
By Pete Croatto
Rambling monologues featuring rhyming dialogue. Lead characters named "He" and "She." Camerawork aching to be lauded in Film Comment. A maid serving as a philosophical voice of reason. It's all there in Yes, Sally Potter's endless, numbing cinematic essay on... on... something.
"She" (Joan Allen) is a London-based scientist (born in Belfast, raised in America) whose open marriage to her stoic, stuffy husband (Sam Neill) is dying a slow, painful death. "He" (Simon Abkarian) is a cook from Beirut, who meets her at a party, beginning a torrid affair that puts both on a physical and emotional trek taking them to Beirut, Belfast, New York, and a groovy Cuba.
Continue reading: Yes Review
Christopher Sheppard Movies
The Party Movie Review
Comedies don't get much darker than this pitch-black British movie, written and directed by Sally...
Ginger and Rosa Movie Review
An extraordinary cast lifts this grim British drama into something watchable, even if the script...
Yes Movie Review
Rambling monologues featuring rhyming dialogue. Lead characters named "He" and "She." Camerawork aching to be...
Filmmakers Index: 0 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z


Unlike most ARM/MIPS systems, where you have a UART console or something to see output from u-boot, if you mess up the firmware on an x86, you'll have a non-communicative brick on your hands.
Of course you can also have firmware issues on ARM/MIPS, if you manage to corrupt u-boot on SPI flash, but since u-boot is open and not proprietary it's easy to rebuild it from source and flash it again to recover.
Not so in the PC world. UEFI is horrendously complex compared to u-boot, and Intel's reference implementation known as TianoCore is usually "improved" by several middlemen before going into the final product.
In this case, we've got TianoCore from Intel, insyde adds some magic fairy dust, this goes to Compal (the ODM of this laptop), and finally Dell (who would probably also claim to add magic fairy dust). In the end there's a lot of proprietary fairy dust floating around, and we probably wouldn't be able to boot the laptop if we just build TianoCore from source.
TianoCore itself is open source and BSD licensed, which is why all the vendors use it. Intel manages porting TianoCore to their new platforms, and since it's BSD licensed, it means that someone like insyde can take the working base from Intel and add their proprietary fairy dust without having to release the modified source code.
ive host access to this region.
And this makes sense. If the system allowed unrestricted write access it would be trivial for some malware to write itself into the system firmware, and then you'd have a persistent rootkit. In my opinion, blocking the ability to read portions of the firmware serves no purpose except to discourage reverse engineering attempts.
Thankfully, there is a method known as Flash Descriptor Security Override Strap which can be used to disable the flash descriptor protection.
Amazingly, this area can be accessed without completely disassembling the laptop. Just taking off the palm rest and keyboard, which is about 8 screws, is enough to access the pins. They're right under the LVDS cable to the display. Thanks, Compal!
As you read in the previous post, I only managed to find an E6320 SPI dump online, and ended up with a laptop that worked-ish. I tried for many hours, but I wasn't able to find any free SPI dumps for the E6230 online. After seeing many forums promising to sell you the firmware for outrageous prices, I finally found one that wanted only 8 złoty ($2) to download the SPI dump for the E6230. Principles be damned, I've wasted enough time trying to get this to work. I paid up and downloaded the files for the 4MB and 8MB chips.
Did I really get a valid firmware for $2? Yes!
So, at this point I'm going to cheat a little: I have a working E6230, but I decided when I started this that I would not touch it with a soldering iron or heat gun. If it ain't broke, don't fix it!
Can I get a full firmware dump from the working E6230?
Intel FPT is a proprietary command-line utility created by Intel for flashing firmware files through the computer's internal SPI flashing interface.
Alright, let's go dump the firmware! First you need to remember to apply the Flash Descriptor Security Override Strap, or the CPU will block your read attempt to the ME region.
However, once we've done that, FPT will allow us to dump regions of flash to a file.
At first I tried to dump just the ME and BIOS, using the A12 firmware from above, I reflashed these regions using the FPT utility from FreeDOS. The verification of the BIOS region failed, and when I rebooted I had a brick again.
But, the FPT tool lets you dump all the regions at once! I dumped the entire flash, but now we've got a 12MB file and I don't know where the split is between the 8MB and 4MB flash chips.
After more searching, I found the Intel ME System Tools, including a utility called Flash Image Tool, which allows you to import the firmware image file created by FPT.
Well, the flash component density matches what's in the laptop, so I guess this is correct.
Writing ROM image file "C:\Users\hmartin\Documents\MESYS\Flash Image Tool\v8.1.10.1286\Build\outimage.bin".
Writing MAP file "C:\Users\hmartin\Documents\MESYS\Flash Image Tool\v8.1.10.1286\Build\outimage.map".
Interesting… outimage(1).bin is exactly 8192KB, and outimage(2).bin is exactly 4096KB. I wrote these two images to the respective chips and put them into the laptop. The moment of truth had arrived.
When you're trying to work through firmware issues, it's really helpful to have the flash chip in a socket. Soldering and desoldering gets really old when you have to do it more than a couple of times.
I'm against the sale of these firmware images. I realize that it takes a non-zero amount of time to get the image, but the whole experience of BIOS forums just leaves you feeling dirty. You have no way to verify before payment that the files they're providing even work. Combine this with the fact that most websites want $10-$20 for the SPI dump, the experience leaves a bad taste in your mouth.
Is it worth it to pay? It depends entirely on your willingness to pay. Is this your only computer? Do you need it working now? How much of your own time are you willing to invest to learn about UEFI firmware? I've probably put 20 hours into this project, and I still don't understand the internals of UEFI.
This entry was posted in Embedded, Security and tagged BIOS, dell, E6230, firmware, intel me, Latitude E6230, management engine, PCH, UEFI on 2017/02/03 by Hal Martin.


Biomch-L > Biomch-L forums > Biomch-L 1988-2010 > Help! Req. Information on biomechanics of the human spine by hipdiseases.
View Full Version : Help! Req. Information on biomechanics of the human spine by hipdiseases.
and dynamics of orthopedic patents.


text:
LQDN au Camp Climat
9 August à 19h00 quadrapéro parisienLe Garage, 60 rue des Orteaux (Paris 20e)
LQDN au CC Camp
Joint Civil-Society Statement Addresses Net Neutrality in Europe
Paris, 2 may 2016 — In a joint letter, 73 organisations from 31 countries call on the European Telecom Regulators to uphold net neutrality in their current negotiations about the future of the Internet in Europe.
After 2 years the EU adopted a net neutrality law which leaves many core questions up for interpretation. The Telecom Single Market regulation was adopted in October 2015 in second reading in the European Parliament.
The Body of European Regulators of Electronic Communication (BEREC) and the 28 telecom regulators are currently negotiating guidelines that clarify the recently adopted ambigous European net neutrality law. The regulators have time until August 2016 to publish their final guidelines and will hold a public consultation in June to July.
Europe is in the final stages of the fight for net neutrality. Whether zero-rating, DPI and paid prioritization are allowed for half a billion people depends on the final guidelines that will be published late August 2016. Europe could either follow the global trend towards strong safeguards or set a dangerous precedent.
"This is important that BEREC's guidelines lay the foundations for a neutral and open internet for the upcoming years. This is the only way to ensure fundamental freedoms" says Agnès de Cornulier, Legal and Policy Analysis Coordinator at La Quadrature du Net.
The letter from 73 NGOs calls on the telecom regulators to consider:
So called "specialised services" which risk becoming paid fast lanes that circumvent all net neutrality safeguards.They should be tightly defined to only cover services which are technically not possible over the open internet;
Interpreting the EU regulation as banning application-specific Zero-Rating because it is a harmful practice that restricts consumer choice, perpetuates less expensive low data volumes and distorts competition;
Traffic management should be as application-agnostic as possible. When telecom companies decide about the priority of data packages this risks discriminating against services, including encrypted traffic, harming user choice.
The legal text can be read to both allow and prohibit paid fast-lanes, zero-rating or privacy-intrusive traffic managment forms like Deep Packet Inspection. Basically the legislator kicked the can down the road and now the unelected regulators have to decide on the future of Europe's Internet. This is essential to keep pressuring the telecom regulators in order to ensure the interests of European internet users and this despite the major failings of the process. Besides this public warning, La Quadrature du Net and the other organizations which signed this letter invite internet users to report violations of net neutrality they could have observed on the website respectmynet.eu and to take part in the public consultation available on the website savetheinternet.eu and this in order to alleviate the weaknesses of this European regulation and to enlight regulators.
Read also the letter of the 73 organizations addressed to the BEREC.
See also EDRI's analysis.


text:
Horizons ETFs to Launch World's First Psychedelics-Focused ETF
TORONTO, Jan. 22, 2021 /CNW/ – Horizons ETFs Management (Canada) Inc. ("Horizons ETFs") is pleased to announce that it has filed its final prospectus to launch the Horizons Psychedelic Stock Index ETF ("PSYK"). Units of PSYK have been conditionally approved for listing by the NEO Exchange ("NEO") and are slated to begin trading on January 27, 2020, under the ticker symbol PSYK:NEO.
PSYK will be the world's first exchange traded fund ("ETF") focused on the emerging psychedelics opportunity led by life science and pharmaceutical companies. A growing body of clinical research has demonstrated the potential use of psychedelic compounds, such as psilocybin and ketamine, as treatment for mental illness, depression, addiction, post-traumatic stress disorder (PTSD), and other medical conditions. This has resulted in a growing number of public companies listed in North America that are focussed on the development of therapeutic solutions using psychedelics, which now provides the breadth of portfolio necessary for an ETF.
PSYK will seek to replicate, to the extent possible and net of expenses, the performance of a market index that is designed to provide exposure to the performance of a basket of North American publicly-listed life sciences companies having significant business activities in, or significant exposure to, the psychedelics industry. PSYK will be using the North American Psychedelics Index (the "Index") as its market index. The Index is a proprietary index owned and operated by Horizons ETFs, and Solactive AG is the independent calculation agent for the Index.
The Index is designed to provide a measure of the performance of North American publicly-listed life sciences companies focused on psychedelic medicines, and other companies with business activities in the psychedelics industry. The Index has been designed to provide diversified exposure to the psychedelics industry and also to be adaptive to the evolution of the industry in a timely fashion.
Subject to certain size and liquidity minimums, generally, a company is deemed to be eligible to be included in the Index if the company is listed on a regulated stock exchange in Canada or in the United States, and is a producer and/or a supplier of psychedelic medicines; a biotechnology company that is engaged in research and development of psychedelic medicines; or a company that is part of the supply chain for, or a distributor of, psychedelics.
"After decades of restrictions, recent policy changes and exemptions in Canada and the United States have allowed for increased research in the therapeutic application of psychedelic compounds, and the potential to create an entirely new marketplace for drugs derived from psychedelics," said Steve Hawkins, President and CEO of Horizons ETFs. "PSYK will give investors exposure to the leading public companies undertaking this important research and development of treatments for the more than 700 million people globally that, according to the World Health Organization, suffer from some sort of mental illness, addiction or eating disorder."
While many of the organizations engaged in psychedelics-based clinical research are small but growing life sciences companies pursuing novel research, there are a few global multinationals that have developed in-market applications, including the world's largest health company, Johnson & Johnson, which received FDA approval for Spravato, a ketamine-based nasal spray for treatment of depression.
"We launched the world's first Cannabis-focused ETF in 2017, the Horizons Marijuana Life Sciences Index ETF (HMMJ), and we see many similarities between that industry in 2017 when it was in its infancy to the psychedelics industry now. We see the potential for significant growth from this new sector like what we have witnessed with the Cannabis industry during the last few years." said Mr. Hawkins. "At Horizons ETFs we strive to be at the forefront of key global transformative investment themes. We believe the opportunities with psychedelics not only provide a compelling investment case, but also the potential to provide life-changing impact for those suffering with mental illness."
PSYK will close the initial offering of units to its designated broker at the close of business on January 26, 2021, prior to its initial listing of units on NEO on January 27, 2021.
For further information about PSYK, please visit: www.HorizonsETFs.com/PSYK
About Horizons ETFs Management (Canada) Inc. (www.HorizonsETFs.com)
Horizons ETFs Management (Canada) Inc. is an innovative financial services company and offers one of the largest suites of exchange traded funds in Canada. The Horizons ETFs product family includes a broadly diversified range of solutions for investors of all experience levels to meet their investment objectives in a variety of market conditions. Horizons ETFs has over $17 billion of assets under management and 93 ETFs listed on major Canadian stock exchanges.
Commissions, management fees and expenses all may be associated with an investment in exchange traded products (the "Horizons Exchange Traded Products") managed by Horizons ETFs Management (Canada) Inc. The Horizons Exchange Traded Products are not guaranteed, their values change frequently and past performance may not be repeated. The prospectus contains important detailed information about the Horizons Exchange Traded Products. Please read the relevant prospectus before investing.
There are risks associated with this product. PSYK is expected to invest in, and indirectly derive revenues from, companies in the psychedelics industry engaged in legal activities involving psychedelic drugs and substances. While the medical and adult use of certain psychedelic drugs and substances are generally prohibited under U.S. federal law, despite this prohibition, a limited number of states have either sought to decriminalize or authorize the medical use of certain psychedelic drugs and substances in limited circumstances. Clinical trials involving psychedelic drugs and substances are, however, permitted, provided they comply with both state and federal laws applicable to such trials. Psychedelic drugs and substances in Canada are primarily regulated under the Controlled Drugs and Substances Act (the "CDSA"), the Food and Drug Act and the regulations promulgated thereunder. The medical use of certain psychedelic drugs and substances remain illegal under Canadian federal law unless discretionary exemptions are granted under the CDSA, while a limited number of other drugs and substances may be prescribed by a health care practitioner to patients under their care. Adult recreational use of psychedelic drugs and substances remains generally prohibited under the CDSA. Commercial activities involving psychedelic drugs and substances are permitted in Canada by parties who hold the required federal regulatory approvals and licences; however, distribution and sales opportunities for psychedelic drugs and substances are heavily restricted at this time. PSYK will passively invest in companies engaged in legal activities involving psychedelic drugs and substances in Canada and the U.S. where applicable laws permit such activities. PSYK will not be directly engaged in the manufacture, importation, possession, use, sale or distribution of psychedelic drugs or substances in either Canada or the U.S.
Certain statements herein may constitute a forward-looking statement, including those identified by the expression "expect" and similar expressions (including grammatical variations thereof). The forward-looking statements are not historical facts but reflect the author's current expectations regarding future results or events. These forward-looking statements are subject to a number of risks and uncertainties that could cause actual results or events to differ materially from current expectations. These and other factors should be considered carefully and readers should not place undue reliance on such forward looking statements. These forward-looking statements are made as of the date hereof and the authors do not undertake to update any forward-looking statement that is contained herein, whether as a result of new information, future events or otherwise, unless required by applicable law.
Neither Horizons ETFs nor Solactive AG shall have any liability, contingent or otherwise, to any person or entity for the quality, accuracy, timeliness and/or completeness of the Index, the data of the Index, or for delays, omissions or interruptions in the delivery of the Index or data related thereto. Neither Horizons ETFs nor Solactive AG makes any warranty, express or implied, as the results to be obtained by any person or entity in connection with any use of the Index, including but not limited to the trading of or investments in PSYK or related to the Index, any data related thereto or any components thereof. Neither Horizons ETFs nor Solactive AG makes any express or implied warranties, and hereby expressly disclaims all warranties of merchantability or fitness for a particular purpose or use with respect to the Index or any data related thereto. Without limiting any of the foregoing, in no event shall Horizons ETFs or Solactive AG have any liability for any special, punitive, indirect, or consequential damages (including lost profits), in connection with any use by any person of the Index, PSYK or any products related thereto, even if notified of the possibility of such damages. Neither Horizons ETFs nor Solactive AG is an advisor as to legal, taxation, accounting, regulatory or financial matters in any jurisdiction, and is not providing any advice as to any such matter.
SOURCE Horizons ETFs Management (Canada) Inc.
For further information: For media inquiries: Contact Jonathan McGuire, External Communications Manager, Horizons ETFs Management (Canada) Inc., (416) 640-2956, jmcguire@horizonsetfs.com
Tags: PSYK
Numinus Announces Grant of Stock Options
Awakn Life Sciences Initiates the First Ketamine Treatment Study for Gambling Addiction Led by Professor Celia Morgan
Numinus to Host Extension of MAPS-Sponsored MDMA-Assisted Therapy for PTSD Trials


text:
NETFLIX & HULU
MOVIE NEWS 2021
FUN PERSONALITY QUIZZES
Trump Claimed Not To Know Lil Jon... But They Were On 'Celebrity Apprentice' Together
Did Trump genuinely forget Grammy award-winning artist and 'Celebrity Apprentice' contestant Lil Jon, or does he just not know what's real?
By Sophy Ziss on
These days, presidential press conferences are few and far between. The few that do happen are sprinkled with easily disprovable lies — few of which are entertaining. But on November 7, Trump claimed he didn't know Celebrity Apprentice contestant Lil Jon. The reality TV star said he'd never met a man who participated on HIS TV show and has been photographed with him on multiple occasions. That's a real thing that happened. Live, on the news!
Here's why this came up at all: A reporter asked the President about allegations that he had used offensive language to describe people of color. Specifically, per the reporter, it was said Trump described the Grammy-winning artist as "Uncle Tom," which has been a slur for more than 150 years. Instead of denying it outright or making an impassioned speech about how he'd never do that, Trump went a different route. He took time from his presidential conference to remark, "I don't know him. I don't know who that is."
First of all, everyone knows who Lil Jon is. Even if you haven't kept up with him since he emerged in the mid-'90s, by the end of 2003, he was inescapable. His influence on music and culture is everlasting. This dude won a Grammy award for the song "Yeah!" in 2005. Everyone knows Lil Jon.
But more than that — so much more than that — Lil Jon was a contestant on The Celebrity Apprentice in 2011. You know, the show hosted and made famous by Donald J. Trump. Cue Lil Jon voice: "Whaaat?"
"I don't know who Lil Jon is," President Trump tells a reporter when asked about a claim made by the rapper about Trump's rhetoric.
"The point of fact is I've never used a racist remark" https://t.co/pwWYYzlnug pic.twitter.com/0e3LQmnOLK
— ABC News Politics (@ABCPolitics) November 7, 2018
The internet was quick to jump on this. Twitter covered it all afternoon as users shared photos of the two standing side by side as well as clips of a red carpet event they attended together. If they'd met in passing at an event or if the rapper had competed in one of the first seasons of the show, Trump's confusion would be understandable. If they'd never met and Lil Jon was just a famous musical artist, it would make sense that Trump wouldn't recall him. But that isn't the case. Lil Jon lasted eight of the 12 episodes on the 2011 season of The Celebrity Apprentice. There's photo *and* video evidence that the two attended at least one publicity event for it together. Come on. Trump's strongest defense against allegations of racist remarks was to say he didn't know the guy making the claims. A moment later, the reporter pointed it out, at which point Trump *seemed* to remember who he was. Whether he actually did is anyone's guess.
Interestingly, a lot of people do know Lil Jon was a Celebrity Apprentice contestant, for the same reason they know Meatloaf and Gary Busey were contestants in the same season. It's not that they watched — it's that then-President Barack Obama brought it up during the 2011 White House Correspondents' Dinner. During that speech, which is kind of shocking to watch now, Obama teases Trump about his presidential aspirations. He literally lays out a scene from Celebrity Apprentice that involves contestants Gary Busey, Meatloaf, and Lil Jon. As the world later learned, Obama had given the go-ahead to take out Osama bin Laden the same evening.
The full speech has over 10 million views on YouTube. There's also a 2016 Washington Post article that gently suggests that night's speech had a hand in causing Trump to run for president for real. Of course, no one can say for sure.
The Trump administration lies all the time, about everything. Now, that includes something as innocuous as Trump's history with Lil Jon. It's just so bizarre.
Can we reset the Matrix simulation now, please?
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X Factor 26th and 27th October Show – Miss Dynamix are voted off!
So this week's X Factor was 'Movie Night' and it looks like Miss Dynamix's DVD player was broken because they lost in a sing off against Hannah.
It all came down to Louis and he decided to send the band home instead of Hannah after Sharon had said it was the "best sing off" of the show.
Who do you think should have left the show? Let us know in the comment box below! Oh, and no, you can't vote off Luke Friend's hair.
We're always playing pur favourite X Factor stars on Fun Kids. Tune in to catch One Direction, Little Mix, Olly Murs and more!